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Mathematical properties of the Platonic solids

One can specify the following mathematical characteristics in each of the five Platonic solids:

1. The radius of the sphere circumscribing the polyhedron;

2. The radius of the sphere inscribed in the polyhedron;

3. The surface area of the polyhedron;

4. The volume of the polyhedron.

Tetrahedron:  All four faces are equilateral triangles.

a circumscribed sphere of tetrahedron

The radius of a circumscribed sphere
of tetrahedron is

Радиус сферы описанной вокруг тетраэдра

 

where a is side length.

an inscribed sphere of tetrahedron

The radius of an inscribed sphere
of tetrahedron is

Радиус сферы вписанной в тетраэдр

The surface area of tetrahedron

The surface area of tetrahedron S tetr
It is possible to represent the surface area of tetrahedron as an area of the model for better illustration.

The volume of tetrahedron

 

The volume of tetrahedron is

V tetr

Octahedron:  All eight faces are equilateral triangles.

a circumscribed sphere of an octahedron

The radius of a circumscribed sphere
of an octahedron is

 

where a is side length.

an inscribed sphere of octahedron

The radius of an inscribed sphere
of octahedron is

The surface area of octahedron

The surface area of octahedron is

It is possible to represent the surface area of octahedron as an area of the model for better illustration.

The volume of octahedron

 

The volume of octahedron is

Hexahedron (cube):  All six faces are squares.

a circumscribed sphere of cube

The radius of a circumscribed sphere
of cube

 

where a is side length.

an inscribed sphere of cube

The radius of an inscribed sphere
of cube is

surface area of cube

The surface area of cube is

 

It is possible to represent the surface area of cube as an area of the model for better illustration.

The volume of cube

 

The volume of cube is

Dodecahedron: All 12 faces are regular pentagons.

a circumscribed sphere of dodecahedron

The radius of a circumscribed sphere
of dodecahedron

 

where a is side length.

an inscribed sphere

The radius of an inscribed sphere
of dodecahedron is

The surface area of dodecahedron

The surface area of dodecahedron is 
 
It is possible to represent the surface area of dodecahedron as an area of the model for better illustration.

Объем додекаэдра

The volume of dodecahedron is

Icosahedron: All 20 faces are equilateral triangles.

a circumscribed sphere of icosahedron

The radius of a circumscribed sphere
of icosahedron

where a is side length.

a circumscribed sphere of icosahedron

The radius of an inscribed sphere
of icosahedron is

The surface area of icosahedron

The surface area of icosahedron is
 
It is possible to represent the surface area of icosahedron as an area of the model for better illustration.

the volume of icosahedron

 

the volume of icosahedron is

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